Formal Derived Algebraic Geometry
Chang-Yeon Chough
TL;DR
This work develops the foundations of formal derived algebraic geometry and builds a bridge to formal spectral geometry by relating derived and spectral Deligne–Mumford stacks through the forgetful functor Theta and its adjoints. It introduces adjunctions between derived and spectral settings at the level of infinity-topoi (via Xi and Xi^R) and demonstrates that affine objects, étale morphisms, and truncations correspond under this bridge, enabling a robust comparison of derived and spectral DM stacks, including their formal completions. The paper then extends these comparisons to the formal/adic setting, showing compatibility of Spf constructions and completions, and defines formal derived DM stacks as colimits of affine formal spectra. As an application, it proves a derived version of formal GAGA: for a complete adic base and appropriate derived spaces X and Y, the mapping spaces before and after formal completion are homotopy equivalent, extending Lurie’s spectral formal GAGA to the derived context. These results provide a cohesive framework connecting derived and spectral geometries and enable transference of coherence and mapping-space phenomena across contexts.
Abstract
We develop some foundations for the theory of formal derived algebraic geometry, which parallel the theory of formal spectral algebraic geometry by Jacob Lurie. For this, we establish a close connection between algebro-geometric objects in the derived and spectral settings. We apply this construction to prove a version of the formal GAGA theorem in the derived setting.